On the approximation of spectra of linear operators on Hilbert spaces

We present several new techniques for approximating spectra of linear operators (not necessarily bounded) on an infinite dimensional, separable Hilbert space. Our approach is to take well known techniques from finite dimensional matrix analysis and show how they can be generalized to an infinite dimensional setting to provide approximations of spectra of elements in a large class of operators. We conclude by proposing a solution to the general problem of approximating the spectrum of an arbitrary bounded operator by introducing the n-pseudospectrum and argue how that can be used as an approximation to the spectrum.